improvement of modeled soil wetness conditions and turbulent

Improvement of Modeled Soil Wetness Conditions and Turbulent Fluxes through theAssimilation of Observed Discharge

VALENTIJN R. N. PAUWELS AND GABRIËLLE J. M. DE LANNOY

Laboratory of Hydrology and Water Management, Ghent University, Ghent, Belgium

(Manuscript received 23 February 2005, in final form 8 August 2005)

ABSTRACT

The objective of this paper is to improve the performance of a hydrologic model through the assimilationof observed discharge. Since an observation of discharge at a certain time is always influenced by thecatchment wetness conditions and meteorology in the past, the assimilation method will have to modify boththe past and present soil wetness conditions. For this purpose, a bias-corrected retrospective ensembleKalman filter has been used as the assimilation algorithm. The assimilation methodology takes into accountbias in the forecast state variables for the calculation of the optimal estimates. A set of twin experiments hasbeen developed, in which it is attempted to correct the model results obtained with erroneous initialconditions and strongly over- and underestimated precipitation data. The results suggest that the assimi-lation of observed discharge can correct for erroneous model initial conditions. When the precipitation usedto force the model is underestimated, the assimilation of observed discharge can reduce the bias in themodeled turbulent fluxes by approximately 50%. This is due to a correction of the modeled soil moisture.In the case of an overestimation of the precipitation, an improvement in the modeled wetness conditions isalso obtained after data assimilation, but this does not lead to a significant improvement in the modeledenergy balance. The results in this paper indicate that there is potential to improve the estimation ofhydrologic states and fluxes through the assimilation of observed discharge data.

1. Introduction

The study of the relationship between precipitationand catchment discharge has always been one of themajor interests in hydrology. Models describing this re-lationship have been developed conceptually, meaningthat the complex physical reality is simplified using spe-cific hypotheses and assumptions (Beven 2000). Thedrawback of such models is that their range of applica-bility is limited to the conditions under which they havebeen developed. To better model the rainfall-runoff be-havior of a catchment, physically based models havebeen developed. More specifically, the goal of this typeof model is to describe the partitioning of the incomingsolar and atmospheric radiation into latent, sensible,and ground heat fluxes, and the partitioning of the pre-cipitation into surface runoff, infiltration, and evapo-transpiration. The study of these processes has led to

the development of soil–vegetation–atmosphere trans-fer schemes (SVATS). A general overview of the over-all performance of a number of these models can befound in Wood et al. (1998).

It is well known that results from SVATS are proneto errors due to a variety of reasons, which can beerrors or oversimplifications in the formulation of themodel physics, and/or errors in the meteorological forc-ing data. Other reasons include the lack of soil, vegeta-tion, and topographic data at a sufficiently high resolu-tion and/or errors in these datasets and the derivedparameters. Remote sensing of surface state variables(e.g., soil temperature, soil moisture) or surface fluxes(e.g., latent heat fluxes) forms an excellent opportunityto update SVATS with external datasets (Troch et al.2003). The update of the model state with externallymeasured variables is commonly referred to as dataassimilation.

A wide variety of studies have put data assimilationinto practice. The most frequently used methods to up-date the soil moisture state of land surface models aredirect insertion (Heathman et al. 2003), Newtoniannudging (Houser et al. 1998; Pauwels et al. 2001; Pani-

Corresponding author address: Valentijn R. N. Pauwels, Labo-ratory of Hydrology and Water Management, Ghent University,Coupure links 653, B-9000 Ghent, Belgium.E-mail: [email protected]

458 J O U R N A L O F H Y D R O M E T E O R O L O G Y VOLUME 7

© 2006 American Meteorological Society

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coni et al. 2002), optimal interpolation (Seuffert et al.2004), Kalman filtering (Galantowicz et al. 1999;Hoeben and Troch 2000; Walker and Houser 2001;Walker et al. 2001; Reichle et al. 2002a,b; Margulis et al.2002; Crow and Wood 2003), Kalman smoothing(Dunne and Entekhabi 2005), and variational data as-similation (Castelli et al. 1999; Reichle et al. 2001a,b;Boni et al. 2001; Caparrini et al. 2003, 2004; Margulisand Entekhabi 2003; Crow and Kustas 2005). Several ofthese studies have used remotely sensed data, includingradar and radiometric data. Recently, remotely sensedlatent heat fluxes have been assimilated into a hydro-logic model in order to update the soil moisture state(Schuurmans et al. 2003). Other studies have focusedon the assimilation of surface temperature (Boni et al.2001; Lakshmi 2000), or have assimilated both the soilmoisture content and surface temperature (Walker etal. 2001).

The application of hydrologic models at large spatialscales renders the use of in situ observations difficultfor the update of the model state. In this case remotesensing data are better fitted for the observation of thesystem state. Although it has been shown that the as-similation of remotely sensed data can lead to an over-all increase in model performance, the use of remotelysensed data suffers from some significant drawbacks.Remote sensing data usually have a larger uncertaintythan in situ observations and are usually available at acoarse temporal (e.g., daily, weekly, or monthly prod-ucts) and spatial (10 m to 50 km) resolution. Because ofthese drawbacks, an alternative to using remote sens-ing observations is the use of catchment discharge ob-servations in order to update the model soil wetnessstate at the catchment scale. Discharge observationsrepresent the integrated effect of point-scale runoff-generating processes and catchment topography, andare an excellent indication of the catchment wetnessconditions. The major advantage of the use of dischargeobservations for the updating of hydrologic models isthat these observations are generally available at a hightemporal resolution. Further, catchment discharge isusually monitored with a reasonable degree of confi-dence (Boiten 2000). Discharge measurements are alsoavailable across different spatial scales, in contradictionto remotely sensed observations of, for example, soilmoisture, and they can thus be used for the improve-ment of model results across different spatial scales.The use of measured river discharge values for the up-dating of the state of hydrologic models through Kal-man filtering has been given a good deal of attention inthe past. Kalman filtering and observed discharge datahave been used to update the state variables and/orparameters of simple linear (mostly time series) (Todini

and Wallis 1978; Wood 1978; Grigoriu 1978; Katayama1980; Bras 1980; Ganendra 1980) and nonlinear rain-fall-runoff models (Logan et al. 1978; Katayama 1980;Bras 1980). Recently, the possibility to improve theforecasts of simple conceptual rainfall-runoff modelshas been indicated by Aubert et al. (2003). In theirstudy, the results from a simple lumped rainfall-runoffmodel, consisting of two reservoirs (a soil and routingreservoir) for the entire catchment, and requiring onlyfour input parameters, have been updated through theassimilation of observed soil moisture values and/or dis-charge records. The objective of all these studies was toimprove runoff forecasts, not the modeled soil moisturestate or energy balance terms.

The overall goal of this paper is to improve the per-formance of a fully process-based lumped water andenergy balance model [the TOPMODEL-Based Land–Atmosphere Transfer Scheme (TOPLATS; Famigliettiand Wood 1994)] through the assimilation of observeddischarge values. More specifically, the first objective ofthis study is to correct the modeled soil wetness condi-tions and turbulent fluxes if the initial conditions arewrong. Further, it is known that precipitation is themost important variable in the determination of the soilwetness conditions, which in turn are very important inthe calculation of evapotranspiration rates. For this rea-son, the second objective of this paper is to assesswhether the assimilation of observed discharge can cor-rect modeled soil moisture values, and consequentlyimprove the modeled energy balance, when the precipi-tation used to force the model is either over- or under-estimated. All model applications are done over theZwalm catchment in Belgium, using hourly observedforcing data from 1994 through 1998.

The paper is organized as follows. First, the assimi-lation algorithm is explained. Then, a short descriptionof the hydrologic model, runoff routing algorithm, testsite, and datasets is given. After this the results of thevarious model applications are analyzed. Finally, a sum-mary of the obtained results and conclusions that canbe drawn from this work are given.

2. The assimilation algorithm

In the ensemble Kalman filter (EnKF) (Evensen1994), the following system is considered:

�Xk�1 � fk�Xk, uk� � wk

Yk � HkXk � �k

, �1�

where Xk is the system state vector with ns entries, Yk

is the system observation vector with no entries, uk isthe system forcing, wk is the process noise, �k is the

JUNE 2006 P A U W E L S A N D D E L A N N O Y 459

measurement noise, and k indicates the time step. Inour case, Xk consists of the soil moisture content (of theupper and lower soil layers) and catchment-averagedwater table level, and Yk consists of the measured dis-charge; Hk relates the system state to the observation ofthis system state (in our case it thus relates the soilwater content and water table level to the discharge)and is calculated numerically. The statistics of wk and �k

can be summarized as follows:

�E�wk� � 0, E��k� � 0

E�wkwkT� � Qk � 0, E��k�k

T� � Rk � 0

E�wkwlT� � 0, E��k�l

T� � 0, �k � l

E�wk�lT� � 0, �k, l

,

�2�

where Qk is the process noise covariance, Rk is themeasurement noise covariance, and T stands for thetranspose operator. Given the above definitions, Eq.(1) is an accurate description of the system for the pur-pose of assimilating data that are, at each measurementtime step, determined by the system state at the sametime step. However, runoff generated at a certain timeand location in a catchment will only reach the catch-ment outlet after a certain time period. For this reason,a runoff observation at a certain time is determined bythe state of a catchment at a number of previous timesteps. Thus, if runoff time series are assimilated into ahydrologic model, not only the state of the catchmentfor a given location at the current time step, but also thestate at a number of different locations and previoustime steps, have to be updated and propagated forwardto the current time step. For this reason, the systemstate vector Xk is augmented so that it contains not onlythe system state at time step k, but also the system stateat time steps k � 1, k � 2, . . . , k � (nc � ny � 1). Herenc is the number of time steps in the concentration timeof the catchment (the number of time steps it takes forsurface runoff generated at the point in the catchmentfarthest from the outlet to reach the outlet), and ny isthe number of time steps with observed discharge val-ues in the observation vector. Using this representation,an observation at time step k will thus be used to up-date the model state at time steps k through k � nc. Ifobservations at time step k and k � 1 are used, themodel state at time steps k through k � nc � 1 will beupdated. These model states at time steps k � i, with iranging from zero to nc, are in fact the initial conditionsfor time step k � i � 1. Using these initial conditions,the model is then reapplied for time steps k throughk � nc � 1, from which the modeled fluxes are calcu-lated. The approach of distributing the system state

vector in time, and the use of observed data to updatethe model state at a number of time steps prior to theobservation time steps, using the traditional EnKFequations, is referred to as retrospective ensemble Kal-man filtering (REnKF).

Figure 1 explains this system representation in moredetail. Assume a catchment has a unit hydrograph thatlasts for 10 time steps. If one wants to assimilate arunoff observation at a certain time step, the systemstate up to 10 time steps before the runoff observationhas to be updated. If, for example, one wants to assimi-late six consecutive runoff observations (this could beany number; in this example the number six was ran-domly chosen), the system state starting 10 time stepsbefore the first observation, up to the state at the timeof the last observation, thus in total 16 time steps, has tobe updated. The size of the assimilation window (thenumber of observations taken into account) can be cho-

FIG. 1. Schematic of the data assimilation algorithm. (top) Thecomparison between the discharge simulations and observations.(middle) The precipitation time series that were used to obtainthe discharge simulations. (bottom) The unit hydrograph of thecatchment.


sen based on computational efficiency, or an optimalvalue can be determined based on a comparison be-tween the results of the assimilation procedure and ob-servations. The objective of this paper is to examine thedata assimilation algorithm with an assimilation win-dow of one observation. An examination of the optimalsize of the assimilation window is the purpose of a fur-ther study.

A number of other methods are available to updatestate variables at a certain time step using observationsfrom different time steps and that potentially could beused for the assimilation of runoff observations, bymaking the observations Y at time step k in Eq. (1)dependent not only on the state X at time step k, butalso at time steps k � 1, k � 2, and so on. For example,Cohn et al. (1994) developed a Kalman smoothing ap-proach, in which the system state vector is not distrib-uted in time, but in which the cross covariances be-tween the errors at the current and the previous timesteps have to be propagated in order to allow the cal-culation of the Kalman gain factors. In this method, adifferent gain factor is calculated for every time step atwhich the model state variables have to be updated.Todling and Cohn (1998) developed and analyzed anumber of simplifications to this fixed-lag Kalmansmoother. Another method to update the system stateat a certain time before the observation was developedby Evensen and van Leeuwen (2000), in which a firstguess for the analysis is the result of the application ofthe ensemble Kalman filter. This first guess is thenpropagated backward in time by using the ensembleerror cross covariances. In other words, every time anew dataset becomes available during the forward in-tegration, an analysis is first computed for that timestep using the ensemble Kalman filter solutions. Sincethe covariances between the states at the observationtime steps and the states at previous time steps areknown, the update at the observation time step can bepropagated backward in time for all time steps up tothat time. The approach of distributing the system statevector in time was recently used by Dunne and En-tekhabi (2005) for the assimilation of surface soil mois-ture in a reanalysis approach. In their study, soil mois-ture observations with a temporal resolution of 3 days,

and an assimilation window of three observations, wereused. They concluded that for the purpose of assimilat-ing soil moisture data their method can yield an im-proved state estimate as compared to traditional filter-ing. Because of these promising results, and the relativesimplicity of implementation, the approach of distrib-uting the system state vector in time is used in thisstudy.

An additional difficulty in the assimilation of dis-charge records in this experiment is the representationof forecast bias. If the meteorological data used to forcethe model are biased, or if certain model parametersare wrong (e.g., the hydraulic conductivity), the mod-eled soil wetness, and consequently the modeled energybalance, will also be biased. If this forecast bias is notadequately represented in the assimilation algorithm,the resulting state estimates will not be optimal. In thispaper, we use the online forecast bias estimation andcorrection with feedback approach described in Deeand Da Silva (1998). In this approach, forecast biasestimate and the state estimate are modeled separately,and observations of the system state are used to opti-mize both estimates.

The data assimilation algorithm works as follows. Toapply the REnKF, an ensemble of N system state vec-tors has to be propagated in parallel. Each system statevector represents one single realization of the possiblemodel trajectories and bias.

First, for each ensemble member i, the state and biasestimates are propagated:

�X̂k

i� � fk�1�X̃k�1i� , uk�1

i �

B̂ki� � B̂k�1

i�

X̃ki� � X̂k

i� � B̂ki�

, �3�

where Bk is the forecast bias, defined as the averagemodeled state variable minus the truth. The hat indi-cates an estimate of the variable, the tilde an unbiasedstate estimate, the superscript � the a priori estimate(forecast, before the assimilation of the observations),and the superscript � the a posteriori estimate (analysis,after the assimilation). The a priori system state vectorX̃i�

k of each ensemble member i is used to calculate thea priori state error covariance as follows:

�Pk

� �1

N � 1Dk

�Dk�T

Dk� � �X̃k

1� � X̃k� , . . . , X̃k

i� � X̃k� , . . . , X̃k

N� � X̃k��

X̃k� �

1N

i�1

N

X̃ki�

, �4�


where P�k is the a priori error covariance matrix. The

assimilation algorithm first updates the bias estimate,after which the state estimate is updated. As explainedby Dee and Da Silva (1998), the forecast error covari-ance S�

k , and the bias prediction error covariance Sb�k ,

can be calculated as

�Sk� � �1 � ��Pk

�

Skb� � �Pk

�, �5�

where is estimated by comparing the root-mean-square error (rmse) between the modeled and observedstate variables to the rmse between the modeled statesvariables with removal of the bias and the observations.Division of the second by the first rmse will provide arough estimate of . This is further explained in section6d. The Kalman gain factor for the bias estimates isthen calculated:

Lk � Skb�Hk

T�HkPk� Hk

T � Rk��1. �6�

The estimate of the bias is updated as follows:

B̂ki� � B̂k

i� � Lk�Yk � HkX̃ki� � �k

i �, �7�

where �ik is a random realization of the measurement

error (Burgers et al. 1998). The a priori error covari-ance is then used to calculate the Kalman gain factor:

Kk � Sk� Hk

T�HkSk� Hk

T � Rk��1. �8�

The a posteriori state vector of each ensemble memberis then calculated as follows:

X̃ki� � X̃k

i� � B̂ki� � Kk�Yk � Hk�X̃k

i� � B̂ki�� k

i �.

�9�

If observations are not available, no update is per-formed. The system state at each time step is simply theaverage of the a posteriori system states of all ensemblemembers.

3. Model description

TOPLATS has as its foundation the concept thatshallow groundwater gradients set up spatial patterns ofsoil moisture that influence infiltration and runoff dur-ing storm events, and evaporation and drainage be-tween storm events. The assumption is made that thesegradients can be estimated from local topography[through a soil–topographic index (Sivapalan et al.1987)]. From this foundation, the model was expandedto include infiltration and resistance-based evaporationprocesses, a surface vegetation layer and a surface en-ergy balance equation with an improved ground heatflux parameterization, and the effect of atmosphericstability on energy fluxes (Famiglietti and Wood 1994;

Peters-Lidard et al. 1997). The model was originallydeveloped to simulate the surface water and energybalance for warm seasons (Famiglietti and Wood 1994;Peters-Lidard et al. 1997). More recently, winter pro-cesses (frozen ground and a snowpack), an improvedwater and energy balance scheme for open-water bod-ies, and a two-layer vegetation parameterization wereadded (Pauwels and Wood 1999a). Application to theZwalm catchment (Pauwels et al. 2001, 2002) and tofield experiments such as the First International Satel-lite Land-Surface Climatology Project (ISLSCP) FieldExperiment (FIFE) (Peters-Lidard et al. 1997) and theBoreal Ecosystem Atmosphere Study (BOREAS)(Pauwels and Wood 1999b, 2000) has shown that themodel can adequately simulate surface energy fluxes,soil temperatures, and soil moisture.

The soil column in divided into a thin upper layer(depth 5 cm) and a lower layer. Water for bare-soilevaporation is supplied by the thin upper layer, whilewater for vegetation transpiration can be supplied byeither layer, depending on the distribution of the rootsin the soil column. This distribution is specified for eachland-cover class as a model parameter. Surface runoff isgenerated using both the saturation and infiltration ex-cess mechanisms. When the rate of net precipitationexceeds the infiltration capacity of the soil [which de-pends on the soil moisture content (Philip 1957)], infil-tration excess runoff is generated. When the capillarylayer above the groundwater table reaches the surface,the net precipitation cannot infiltrate and is consideredto be saturation excess runoff. Baseflow is calculatedusing the TOPMODEL approach (Beven and Kirkby1979), which states that the baseflow varies exponen-tially with the depth to the water table. The total catch-ment discharge is then the sum of the baseflow and thesaturation and infiltration excess runoff. For a morecomplete description of the model physics at the pointscale we refer to Pauwels et al. (2002).

Horizontal transport of water is implicitly modeledthrough the TOPMODEL formulation. At the end ofeach time step the sum of the fluxes to and from thewater table are calculated, after which the new aver-age water table depth over the catchment is calculated.This catchment-average water table depth is thenused to calculate the local water table depth usingTOPMODEL (Sivapalan et al. 1987).

While the model can be run in two different modes(distributed and statistical), the statistical mode is usedin this study. The statistical mode has been developedunder the similarity concept that locations within thedomain (or macroscale grid), with the same soil–topographic index and vegetation, respond similarly.Using the statistical distributions of these controlling


factors, rather than applying the model for each grid,considerably reduces the computational effort.

4. The runoff routing algorithm

The runoff routing algorithm was developed byTroch et al. (1994). The algorithm is based on a solutionof the de Saint Venant equations, neglecting the accel-eration terms. When the upstream boundary conditionis given by the Dirac function, Brutsaert (1973) solvedthese equations as follows:

q�x, t� �x

�2�bt3�2exp��

�x � at�2

2b2t�, �10�

where q is the discharge per unit width of channel (s�1),x is the distance along the channel (m), t is the time (s),a is the drift velocity (m s�1), and b2 is the diffusioncoefficient (m2 s�1); a and b are calculated as

�a � �1 � a0�V

b2 �V3

gS0F2 �1 � a02F2�

, �11�

where V is the vertically averaged channel velocity(m s�1), S0 is the channel bed slope (�), F is the di-mensionless Froude number, g is the gravity constant(9.81 m s�1), and a0 is equal to 2/3.

Equation (10) describes the response in the channelat a distance x from the origin of the instantaneous unitperturbation. Using the normalized width function, theresponse of the entire channel network can be ex-pressed as (Mesa and Mifflin 1986)

fc�t� � �0

�

qc�x, t�W�x� dx. �12�

The normalized width function W(x) (m�1) representsthe distribution of runoff entering the network at a dis-tance x from the outlet and can be written as

W�x� �1

LTNc�x�, �13�

where Nc(x) is the number of channel links at a givendistance from the outlet and LT is the total channellength (m).

Troch et al. (1994) expanded this method to calculatethe channel flow to calculate the overland flow. Insteadof calculating the distribution of runoff entering thenetwork at a distance x, the distribution of runoff gen-erated at a distance x from the channel network is cal-culated. This distribution is represented by the normal-ized hillslope function H(x) (m�1):

H�x� �INx

P2 , �14�

where � is the resolution at which the normalized hill-slope function is calculated (m), I is the number of cellsin the channel, and P is the number of cells in thehillslope; Nx is calculated as

Nx �1I i�1

I

Nx�i�, �15�

where Nx(i) is the number of cells that drain directly tocell i at an overland flow distance x. Using Eq. (10) withadapted parameters a and b for the overland flow, andEq. (12) with H(x) instead of W(x), the response of thehillslopes fh(t) (s�1) to an instantaneous unit input ofwater can be calculated.

The response of the catchment to an instantaneousunit input of water [ fb(t), s�1] can then be calculatedthrough convolution:

fb�t��0

�

fh�t � �fc�� d. �16�

The channel and overland parameters were optimizedthrough a study of the time lag between the simulatedand the observed discharge. Figure 2 shows the normal-ized hillslope and width functions, and the resultingunit hydrograph for the Zwalm catchment.

5. Site and data description

Figure 3 shows the location of the Zwalm catchment.For a complete description of the Zwalm catchment werefer to Troch et al. (1993) as only a short overview willbe given here. The total drainage area of the catchmentis 114 km2 and the total length of the perennial chan-nels is 177 km. The maximum elevation difference is150 m. The average year temperature is 10°C, withJanuary the coldest month (mean temperature 3°C)and July the warmest month (mean temperature 18°C).The average yearly rainfall is 775 mm and is distributedevenly throughout the year. The annual actual evapo-ration is approximately 450 mm.

Meteorological forcing data with an hourly resolu-tion (the model time step) from 1994 through 1998 areused in this study. These data were prepared based ondaily observations of air temperature and humidity, so-lar radiation, wind speed, and precipitation, from theclimatological station located in Kruishoutem, Belgium,approximately 5 km outside the catchment (KMI 1994–1998). Pauwels et al. (2002) give a detailed descriptionof the processing of the meteorological forcings. Dis-charge observations at the outlet of the catchment, usedfor the validation of the model, were recorded at anhourly time step. Finally, a 30-m-resolution digital el-evation model (DEM), a soil texture map from the Bel-


gian National Geographic Institute, and a Systeme pourl’Observation de la Terre (SPOT)-derived land coverclassification map from 3 August 1998, were used todefine model parameters. Soil parameters were derivedusing the relationship with soil texture given by Rawlset al. (1982), and the vegetation parameters were de-rived for each land-cover classification following Pe-ters-Lidard et al. (1997). The catchment base flow pa-rameters (the base flow at saturation and the paramaterof the exponential decay in the saturated hydraulic con-

ductivity with depth) were taken from Troch et al.(1993).

6. Results of the model applications

a. Validation of the model

A model run is first established, in which the mod-eled discharge is compared to the observations. Themodel is run from 1994 through 1998, at an hourly timestep. The distribution of the topographic indices con-

FIG. 2. The runoff routing functions for the Zwalm catchment. (top) The normalized widthfunction. (middle) The normalized hillslope function. (bottom) The resulting unit hydrograph.


sists of 13 intervals. Figure 4 shows the comparisonbetween the measured and simulated discharge for theentire simulation period. The average modeled dis-charge is 1.47 m3 s�1 as compared to 1.52 m3 s�1 for theobservations. The correlation coefficient between theobservations and the simulations is 0.80. These statis-tics, together with the comparison in Fig. 4, indicatethat the model is capable of representing the rainfall-runoff dynamics of the Zwalm catchment.

b. Description of the twin experiments

In the validation of the assimilation algorithm obser-vations of the energy and water balance variables areneeded. Although some observations of, for example,soil moisture content, evapotranspiration, and water

table levels were made during the study period, theseobservations have not been done with a sufficient fre-quency and spatial resolution to be useful for a 5-yrmodel validation. For this reason a twin experiment isdeveloped, in which the routed discharge from themodel run described in section 6a is used as observationof the catchment discharge. For the remainder of thispaper, we refer to this discharge as the syntheticallyobserved discharge. The modeled energy and waterbalance terms from this model application are used asthe observations of these terms, and are for the remain-der of this paper referred to as synthetical data.

The ensemble members are generated following theapproach of Reichle et al. (2002a). The meteorologicalforcings are perturbed by adding Gaussian white noise

FIG. 3. The location of the Zwalm catchment.


to the observed meteorological data used to force eachensemble member. The standard deviation of the noiseis 5 K for the air and dewpoint temperatures, 1 m s�1

for the wind speed, 50 W m�2 for the shortwave radia-

tion, 25 W m�2 for the longwave radiation, 10 mbar forthe surface pressure, and 50% of the magnitude for theprecipitation. Furthermore, the saturated hydraulicconductivity, the exponential decay parameter of the

FIG. 4. Comparison between the modeled and simulated discharge for the entire study period.Observations are in solid lines and simulations are in dashed lines.


hydraulic conductivity with depth, and the catchment-averaged base flow at saturation are also perturbedwith a random number with mean zero and a standarddeviation of 50% of their magnitude. The additionalperturbation is based on De Lannoy et al. (2006) inwhich it is shown that a perturbation of only the me-teorological forcings will lead to an unrealistic en-semble spread, and consequently an unrealistic errorcovariance. The initial conditions for the different en-semble members were not disturbed. The number ofensemble members is set to the number of entries in thesystem state vector, in this case 1570. This number isobtained by multiplying 6 (the number of land coverclasses) by 13 (the number of intervals in the distribu-tion of the soil–topographic index), by 2 (both the up-per- and lower-layer soil water content), adding 1 (thecatchment-averaged water table depth) to this number,and then multiplying this number by 10 (the number oftime steps in the unit hydrograph). Only the catchment-averaged water table depth needs to be updated, sincethe TOPMODEL approach calculates the local watertable depth using the catchment-averaged water tabledepth and the topographic indices. Therefore, an up-date of the average water table depth will be distributedto an update of the local water table depth through theTOPMODEL formulations.

Four different experiments are run. In the first twoexperiments, the capability of the data assimilation al-gorithm to correct for erroneous initial conditions isassessed. In the first experiment, the initial water tablelevel is lowered by 0.3 m, and the synthetically observeddischarge from section 6a is assimilated into the model.The impact of the assimilation on the modeled watertable levels and soil moisture contents is assessed. Inthe second experiment, the initial water table is raisedby 0.3 m, and again the synthetically observed dischargeis assimilated. It should be noted that the change in theinitial water table depth will automatically lead to achange in the initial soil moisture values, since these arecalculated based on the assumption of equilibrium ini-tial conditions and the water table depth.

In the third and fourth experiment, the possibility ofthe data assimilation algorithm to correct the modelstate under erroneous model forcings is assessed. Forthis purpose, the precipitation used to force each en-semble member is biased. In the third experiment, theobserved precipitation is multiplied by a random num-ber with mean 1.5 and standard deviation 0.5. In thefourth experiment, the mean and standard deviation ofthis random number are both 0.5. The impact of thedata assimilation on the modeled soil water content andwater table level, and the modeled surface turbulentfluxes, is assessed. In this case there are thus two per-

turbations of the forcing data: the first perturbationconsists of the multiplication of the observed precipita-tion by a random number with average 1.5 or 0.5, andthe second perturbation consists of the addition of arandom number to all the forcings for each ensemblemember, as described in the second paragraph of thissection.

The application of the filter equations requiresknowledge about the statistics of the observation error,�k. Since the objective of this study is to assess thepotential improvement in hydrologic model resultsthrough the assimilation of observed discharge, as afirst step the observations are assumed to be perfect, soa standard deviation of zero is used. All elements in Rk

are thus zero. A sensitivity analysis is then done, inwhich the observation error is modified between 0.01and 0.5 m3 s�1. The improvement in the modeled en-ergy balance as a consequence of the assimilation oferroneous discharge data is then assessed. The mea-surement noise is assumed to be constant throughoutthe simulation.

c. Correction for erroneous initial conditions

If a hydrologic model is forced with correct meteo-rological data, and if the model formulations and pa-rameters are correct, after a certain time period an er-ror in the model initial conditions will disappear. Thus,under these conditions, one can expect the long-termmodel results to be unbiased. For this reason, the pa-rameter and initial estimate of the bias in the assimi-lation algorithm are both set to zero, for the first andsecond twin experiments. Figure 5 shows the impact ofthe data assimilation on the modeled soil wetness con-ditions for the model applications with erroneous initialconditions. Both the initially lowered and raised watertables are corrected quickly (after approximately oneday). The upper-layer soil moisture is, in both cases,also corrected after approximately one day. The mod-eled lower-layer soil moisture is strongly improved bythe assimilation algorithm, but becomes equal to thesynthetical observations only after approximately 50days. Table 1 shows the impact of the corrected soilwetness conditions on the modeled discharge. In bothcases the bias and rmse between the synthetical obser-vations and the model results are basically eliminated.Together with the regression line closer to the 1:1 lineand the increase in the correlation for the REnKF runs,this shows that, as can be expected, the assimilationprocedure leads to a strong improvement in the mod-eled discharge. From these two twin experiments wecan conclude that the assimilation of observed dis-charge can correct for erroneous initial conditions.


d. Correction of results under erroneousprecipitation

It is undisputed that precipitation is the most impor-tant variable in the determination of discharge and soil

wetness conditions, and thus also the modeled evapo-transpiration. On the other hand, it is also widelyknown that this variable is very difficult to observe ac-curately over large spatial scales, because of its highintermittency and strong spatial variability. Because of

FIG. 5. Correction of erroneous initial conditions through the assimilation of syntheticallyobserved discharge. The synthetical observations are in thick solid lines, the results of thebaseline runs are in thin solid lines, and the results of the assimilation runs are in dotted lines.UL stands for upper layer and LL for lower layer. If invisible, the dotted lines coincide withthe thick solid lines.


these two reasons, estimates of precipitation over largespatial scales will often be biased. One of the objectivesof this paper is to check whether the assimilation ofobserved discharge can correct erroneous model resultsobtained under biased precipitation.

The parameter in Eq. (5) is calculated by dividingthe rmse between the modeled and synthetically ob-served state variables with removal of the bias (thussubtracting the bias from the modeled state) by thermse between the model results and synthetical obser-vations without bias removal. The average value for thethree types of state variables (water table level, andupper- and lower-layer water content) and both twinexperiments is approximately 0.6. This value was foundto be similar when it was calculated for 1994 only or forthe entire simulation period. For the remainder of thissection, a value of 0.6 for is used. In section 6f thesensitivity of the model results to the value for isexamined.

Figure 6 shows the impact of the assimilation proce-dure on the modeled water table levels, for the modelapplications in which the precipitation was over- andunderestimated. Results are shown for 1994 only; forthe other years similar results were obtained. One canimmediately see that the assimilation leads to a largertemporal variability in the catchment-averaged watertable level. These peaks can be explained by the waythe precipitation data are perturbed. Since the precipi-tation is multiplied by a random number (with mean 1.5or 0.5), the precipitation and synthetically observed dis-charge will, during periods with nonzero precipitation,be inconsistent with each other. When the precipitationis overestimated, the assimilation algorithm will lowerthe water table, thus reducing the amount of base flowand the amount of saturated areas (thus reducing theamount of surface runoff). The modeled soil moisturewill also be decreased, in order to increase the infiltra-tion and reduce the amount of surface runoff. Thelower amount of base flow and surface runoff will then

assure that the modeled discharge is equal to the syn-thetical observations. In case the precipitation is under-estimated, the opposite will occur. The correction of themodeled discharge, however, is limited, since, in thecase of underestimated precipitation, the amount ofbase flow the model can generate is limited to the baseflow at saturation, and the amount of surface runoff isalways limited to the amount of precipitation. On theother hand, if the precipitation is overestimated, themodel can only increase the amount of infiltration tothe maximum allowed for the soil type and lower thewater table to a level where the base flow becomesinsignificant.

During periods without precipitation, the assimila-tion algorithm will set the water table level equal to thesynthetic data, correcting the base flow, and thus thetotal amount of discharge, since under these conditionsno surface runoff occurs.

Table 1 shows that, for both model applications, thebias in the modeled discharge is almost eliminated. Thermse between the synthetically observed and modeleddischarge is strongly reduced, and the correlation be-tween the synthetically observed and modeled dis-charge is increased.

Figure 6 shows the impact of the assimilation proce-dure on the modeled upper-layer soil moisture. Again,results are shown for 1994 only given that similar resultswere obtained for the other years. In both cases theerrors in the modeled soil moisture are basically elimi-nated, although the impact of the errors in the precipi-tation, and the consequent improvement in the mod-eled soil moisture, is stronger when the precipitation isunderestimated. The stronger impact of an underesti-mation of the precipitation on the modeled soil mois-ture can be explained by the combined effect of evapo-transpiration and infiltration. Table 2 explains this ef-fect. Using the correct precipitation, approximately 500mm of water infiltrates into the soil per year. When theprecipitation is overestimated, the infiltration is in-

TABLE 1. Results of the data assimilation on the modeled discharge. The synthetic observations are the independent variable (x axis),and the model applications are the dependent variable ( y axis). Units are in m3 s�1. Application 1 is the model application with aninitially too deep water table, application 2 is the model application with an initially too shallow water table, application 3 is the modelapplication with overestimated precipitation, and application 4 is the model application with underestimated precipitation. The resultsfor applications 1 and 2 are for 1994 only; the results for the other model applications are for 1994 through 1998.

Application Run X Y Slope Intercept Rmse R

1 Base 1.18 0.96 0.90 �0.10 0.41 0.961 REnKF 1.18 1.18 1.00 �0.00 0.03 1.002 Base 1.18 1.37 0.97 0.22 0.29 0.982 REnKF 1.18 1.18 1.00 0.00 0.02 1.003 Base 1.53 2.55 1.91 �0.36 2.31 0.883 REnKF 1.53 1.68 1.30 �0.31 1.14 0.894 Base 1.53 0.61 0.27 0.20 1.47 0.794 REnKF 1.53 1.53 0.75 0.39 0.61 0.92


creased to approximately 700 mm yr�1, and a reductionin the precipitation leads to a reduction of the infiltra-tion to approximately 285 mm yr�1. If the amount ofsoil water that is discharged as base flow is subtractedfrom the infiltration, approximately 215 mm yr�1 re-

mains available for transpiration when the precipitationis correct. When the precipitation is overestimated, ap-proximately 300 mm yr�1 remains available, and whenthe precipitation is underestimated, approximately 150mm yr�1remains. When the precipitation is overesti-

FIG. 6. Impact of the data assimilation on the modeled water balance variables in case oferroneous meteorological forcings. The synthetical observations are in thick solid lines, theresults of the baseline runs are in thin solid lines, and the results of the assimilation runs arein dotted lines. UL stands for upper layer and LL for lower layer.


mated, the amount of soil water available for root up-take is thus increased by approximately 85 mm, andwhen the precipitation is underestimated, this amountis reduced by approximately 150 mm yr�1. Since in allcases the evapotranspiration remains below the poten-tial evapotranspiration, the effect on the modeled soilmoisture will thus be more pronounced when the pre-cipitation is underestimated.

Figure 6 shows the impact of the assimilation proce-dure on the modeled lower layer soil moisture, againfor 1994 only. The same conclusions, as can be ex-pected, can be drawn for the lower-layer soil moistureas for the upper-layer soil moisture: in both cases theerror in the modeled soil moisture is reduced, but theeffect of the errors in the precipitation, and the conse-quent improvement through data assimilation, is morepronounced when the precipitation is underestimated.Figure 6 shows that after the data assimilation a highererror in the modeled lower-layer soil moisture remainsthan for the modeled upper-layer soil moisture.

Table 2 shows that, when the precipitation is overes-timated, the reduction in the upper-layer soil moistureand the increase in the water table depth lead to anincrease in the infiltration, a reduction in the surfacerunoff, and a reduction in the base flow. For the case ofunderestimated precipitation, the opposite occurs. Theimprovement in the modeled discharge is thus due to animprovement in both the modeled base flow and sur-face runoff.

Figure 7 shows the bias and the standard deviation ofthe noise of the estimated water table levels. For bothmodel applications the bias tends to be rather low inmagnitude, rarely exceeding 5 cm. An overestimationof the precipitation leads to a slightly negative bias, thusindicating a small underestimation of the water tabledepth. An underestimation of the precipitation leads tothe opposite effect. Figure 7 further shows that the er-ror in the estimation of the water level is relatively low(rarely exceeding 10 cm), and that an overestimation ofthe precipitation leads to a larger uncertainty in the

modeled water table level than an underestimation. Aseasonal cycle in the error can also be observed, whichcan be attributed to the seasonal cycle in the evapo-transpiration. As evapotranspiration is very dependenton the amount of precipitation, an error in the precipi-tation will thus have its highest effect during the sum-mer, because the evapotranspiration rates are highestin the summer. Similar results were obtained for thebias and noise of the upper- and lower-layer soil mois-ture contents.

Table 3 shows the results of the assimilation proce-dure on the modeled energy balance terms. When theprecipitation is overestimated, the latent heat flux isslightly increased by the increase in precipitation, andthe data assimilation has a negligible impact on themodeled latent heat flux (or evapotranspiration), aswell as on the other energy balance terms. Togetherwith Table 1, Fig. 6 leads to the conclusion that theincrease in precipitation is partitioned mostly into anincrease in the modeled discharge and not the modeledevapotranspiration, and that this can be explained bythe limited sensitivity of the modeled soil moisture tothe increase in precipitation.

When the precipitation is underestimated, an im-provement in the modeled latent heat flux (or evapo-transpiration) and sensible heat flux can be observed,while the net radiation and ground heat flux are essen-tially insensitive to the data assimilation procedure.The improvement in the modeled turbulent fluxes canbe explained by the improvement in the modeled soilmoisture, as shown in Fig. 6. The underestimation ofthe precipitation is thus partitioned not only into anunderestimation of the discharge, but also an underes-timation of the evapotranspiration. Because the dataassimilation procedure does not fully correct the under-estimation of the lower-layer soil moisture content, themodeled latent heat fluxes will still be underestimatedwhen discharge data are assimilated, but a reduction inthe bias of more than 50% is nevertheless obtained.

Figure 8 shows the impact of the assimilation of the

TABLE 2. Partitioning of the precipitation for the different model applications (mm yr�1).

Variable Synthetic data

Baseline runoverestimatedprecipitation

Assimilation runoverestimatedprecipitation

Baseline rununderestimated

precipitation

Assimilation rununderestimated

precipitation

Precipitation 795.72 1182.47 1182.47 430.62 430.62Infiltration 2.66 13.13 17.64 0.25 0.22Excess runoffSaturation 131.29 265.63 162.38 37.71 84.05Excess runoffSurface runoff 133.95 278.76 180.02 37.96 84.27Baseflow 286.95 407.82 273.08 136.41 353.22Total discharge 420.90 686.58 453.10 174.37 437.49Infiltration 501.88 707.41 825.89 286.16 230.43


synthetic discharge data on the monthly average diur-nal cycles of the modeled latent heat fluxes. Here it canbe seen that the largest improvement occurs when theprecipitation is underestimated, and that this improve-ment is the weakest during the winter. During the otherseasons the improvement is relatively constant through-

out the year. Figure 9 shows these diurnal cycles for themodeled sensible heat flux. The same conclusions canbe drawn as for the latent heat flux.

As a summary, the results of the four twin experi-ments suggest that there is potential to improve theresults of hydrologic models, but that this improvement

FIG. 7. Bias and std dev of the error in the modeled water table levels in case of erroneousmeteorological forcings.


is dependent on the meteorological conditions duringthe model application.

e. Sensitivity to observation error

To more realistically assess the effect of the assimi-lation of observed discharge, a Gaussianly distributederror was added to the synthetical discharge observa-tions. An error of 0.01, 0.1, and 0.5 m3 s�1 was assumed.The effect of this model error on the modeled energybalance was assessed.

Table 3 shows the effect of this observation error. Itcan clearly be seen, as can be expected, that the in-crease in the observation error leads to a decrease inthe accuracy of the modeled discharge. However, thisdoes not have a strong impact on the modeled energybalance. When the precipitation is overestimated, evenan error of 0.5 m3 s�1 leads to a similar improvement asperfect observations. When the precipitation is under-estimated, the accuracy in the modeled energy balanceslightly increases as the observation error decreasesfrom 0.5 to 0.1 m3 s�1, but a further decrease in theobservation error does not lead to a further improve-ment in the model results. From this sensitivity analysiswe conclude that even under considerable errors in thesynthetic observations (for the Zwalm catchment anerror of 0.1 m3 s�1 is a relatively high error), thereremains a potential to improve the modeled water andenergy balance terms through the assimilation of dis-charge data.

f. Sensitivity to the estimate of

In many cases it will be impossible to make an accu-rate estimate of , more specifically if no observations

of soil moisture and/or water table levels are available.The objective of this section is to assess whether theassimilation algorithm will also provide acceptable re-sults if the bias correction algorithm is omitted, in otherwords if is set to zero. For these model applicationsthe observation error was also set to zero. Table 3shows the results of these model applications, for thecases in which the precipitation was over- and under-estimated. When the precipitation is overestimated, thebias between the model results and the synthetic obser-vations is slightly larger as compared to the model runswhere is set to 0.6, but the rmse is slightly lower.When the precipitation is underestimated the oppositebehavior can be observed. Since the differences be-tween the model results with zero and 0.6 are so low,it can be concluded that, when it is impossible to esti-mate , a value of 0 will lead to results that will besimilar to the results with an accurate estimate , eventhough these results will not be accurate estimates.

7. Summary and conclusions

A method has been developed to update the paststate of a land surface model using discharge observa-tions in a retrospective ensemble Kalman filter frame-work. The method allows the correction of bias in themodel forecasts. For each model state update, one dis-charge observation is used, although it is possible to useany number of discharge observations for the updatingof the model. In a twin experiment it is shown thaterroneous initial conditions can be corrected forthrough the assimilation of synthetically observed dis-charge. A second set of twin experiments is then devel-

TABLE 3. Results of the data assimilation on the modeled energy balance parameters for the model run with overestimated precipi-tation. Here, Rn is the modeled net radiation; LE, H, and G are the modeled latent, sensible, and ground heat fluxes, respectively; ETis the modeled evapotranspiration; and Q is the modeled discharge. The synthetic observations are the independent variable (x axis);the model applications are the dependent variable ( y axis). Units are in W m�2, except for the evapotranspiration, which is in mm perhour, and for the discharge, which is in m3 s�1. The averages for the evapotranspiration are recalculated to mm per year. The top partof the table is for overestimated precipitation, and the bottom part for underestimated precipitation.

X Baseline run Error 0.00 m3 s�1 Error 0.01 m3 s�1 Error 0.10 m3 s�1 Error 0.50 m3 s�1 0

Y rmse Y rmse Y rmse Y rmse Y rmse Y rmseRn 96.88 97.26 1.00 97.11 0.69 97.11 0.69 97.11 0.68 97.09 0.71 97.09 0.65LE 45.08 48.05 6.20 47.17 4.77 47.16 4.75 47.12 4.62 47.06 4.61 47.04 4.50ET 336.73 374.98 0.0093 363.84 0.0071 363.71 0.0070 363.11 0.0069 362.32 0.0068 362.13 0.0067H 47.12 44.33 6.80 45.15 4.71 45.16 4.69 45.20 4.58 45.27 4.63 45.28 4.40G 4.68 4.88 2.58 4.79 1.65 4.79 1.64 4.79 1.65 4.76 1.85 4.77 1.58Q 1.53 2.55 2.30 1.68 1.14 1.71 1.19 1.84 1.63 1.92 1.39 1.66 1.35Rn 96.88 96.04 1.98 96.44 1.32 96.44 1.32 96.47 1.34 96.68 1.47 96.44 1.34LE 45.08 38.36 13.30 41.67 9.28 41.68 9.28 41.88 9.32 43.21 9.40 41.58 9.48ET 336.73 250.27 0.0197 293.10 0.0136 293.14 0.0136 295.83 0.0137 313.06 0.0138 291.92 0.0139H 47.12 53.39 13.74 50.33 8.42 50.33 8.42 50.13 8.49 48.86 8.81 50.43 8.64G 4.68 4.29 4.91 4.43 3.32 4.43 3.31 4.45 3.39 4.61 4.01 4.43 3.31Q 1.53 0.61 1.47 1.53 0.61 1.53 0.62 1.64 0.73 2.09 1.14 1.48 0.60


FIG. 8. Impact of the data assimilation on the monthly average diurnal cycle of the modeled latent heatfluxes (LE). The monthly averages are calculated for the 5-yr simulation period. The double-thick solidlines are the synthetic observations, the thick solid lines are the results of the baseline run with under-estimated precipitation, and the thick dotted lines are the results of the baseline run with overestimatedprecipitation. The thin solid lines are the results of the baseline run with underestimated precipitation,and the thin dotted lines are the results of the baseline run with overestimated precipitation.


FIG. 9. Impact of the data assimilation on the monthly average diurnal cycle of the modeled sensibleheat fluxes (H ). The monthly averages are calculated for the 5-yr simulation period. The double-thicksolid lines are the synthetic observations, the thick solid lines are the results of the baseline run withunderestimated precipitation, and the thick dotted lines are the results of the baseline run with over-estimated precipitation. The thin solid lines are the results of the baseline run with underestimatedprecipitation, and the thin dotted lines are the results of the baseline run with overestimated precipita-tion.


oped, in which the precipitation is first strongly over-estimated, and then strongly underestimated. A reduc-tion in the bias between the synthetically observed andmodeled soil wetness conditions and fluxes is obtainedthrough the assimilation procedure. When the precipi-tation is underestimated, these improved soil wetnessconditions lead to an improvement in the modeled en-ergy balance. The results have been found to be insen-sitive to the error in the observed discharge data, and tothe estimate of the bias parameter . The results indi-cate that it is possible to improve the performance ofland surface models through the assimilation of ob-served discharge records. Further research will focus onthe determination of the optimal size of the assimilationwindow and on the application of the assimilationmethod for forecasting purposes.

Acknowledgments. The lead author was, during thecourse of this work, a postdoctoral researcher fundedby the Institute for Scientific Research of the FlemishCommunity (FWO-Vlaanderen). The second authorwas supported by Research Grant BOF 011D2901 fromthe Special Research Fund (BOF) of Ghent University.This research was also partially funded throughSTEREO Grant SR/00/01 of the Belgian Science Policy(BELSPO). We would also like to thank three anony-mous reviewers and the guest editor of this journal fora thoughtful and constructive review.

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improvement of modeled soil wetness conditions and turbulent

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